Finite Group Databasebeta

← #10608841

Subgroups — #10608841

Order 640.

814 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4083C211yesmaximal, characteristic
2C2×#4029C211yesmaximal, characteristic, complemented
3C2×D5×C22⋊C4C211yesmaximal, characteristic, complemented
4C2×D5×C22⋊C4C211yesmaximal, characteristic, complemented
5C5×#394C211yesmaximal, characteristic, complemented
6#65511C211yesmaximal, characteristic, complemented
7#65278C211yesmaximal, characteristic, complemented
8#65276C211yesmaximal, characteristic, complemented
9#65245C211yesmaximal, characteristic, complemented
10#65245C211yesmaximal, characteristic, complemented
11#65245C211yesmaximal, characteristic, complemented
12#65242C211yesmaximal, characteristic, complemented
13#65241C211yesmaximal, characteristic, complemented
14#64956C211yesmaximal, characteristic, complemented
15#64953C211yesmaximal, characteristic, complemented
16C24×D5C2211yescharacteristic
17C22×C5×D4C2211yescharacteristic
18C22×C22⋊D5C2211yescharacteristic
19C22×C22⋊D5C2211yescharacteristic
20C22×C4×D5C2211yescharacteristic
21C22×C4×D5C2211yescharacteristic
22C22×C4×D5C2211yescharacteristic
23C22×C4×D5C2211yescharacteristic
24C2×C5×C4⋊C4C2211yescharacteristic, complemented
25C2×C5×C22⋊C4C2211yescharacteristic
26C2×C5×C22⋊C4C2211yescharacteristic
27C2×C5×C22⋊C4C2211yescharacteristic, complemented
28C2×C5×C22⋊C4C2211yescharacteristic, complemented
29C2×C22⋊Dic5C2211yescharacteristic
30C2×C22⋊Dic5C2211yescharacteristic
31C2×C22⋊Dic5C2211yescharacteristic
32C2×C22⋊Dic5C2211yescharacteristic, complemented
33C2×C22⋊Dic5C2211yescharacteristic, complemented
34#4083—24no—
35C2×#685C2211yescharacteristic
36C2×#685C2211yescharacteristic
37C2×#685C2211yescharacteristic
38C2×#685C2211yescharacteristic
39C2×#685C2211yescharacteristic
40C2×#685C2211yescharacteristic
41C2×C4⋊Dic5C2211yescharacteristic, complemented
42C2×#683C2211yescharacteristic, complemented
43C2×#683C2211yescharacteristic, complemented
44#4029—24no—
45D5×C22⋊C4—24no—
46D5×C22⋊C4—24no—
47C5×#95C2211yescharacteristic, complemented
48#3963C2211yescharacteristic, complemented
49#3963C2211yescharacteristic, complemented
50#3963C2211yescharacteristic, complemented
51#3963C2211yescharacteristic, complemented
52#3963C2211yescharacteristic, complemented
53#3963C2211yescharacteristic, complemented
54#3963C2211yescharacteristic, complemented
55#2412—51nomaximal, Sylow
56C24×C5C2311yescharacteristic
57C24×C5C2311yescharacteristic
58C23×D5—22no—
59C23×D5—22no—
60C23×D5—22no—
61C23×D5—22no—
62C23×D5—41no—
63C23×D5—41no—
64C23×D5D414yes—
65C23×D5C2311yescharacteristic
66C23×D5C2311yescharacteristic
67C2×C5×D4—22no—
68C2×C5×D4—41no—
69C2×C5×D4—41no—
70C22×C4×C5C2311yescharacteristic
71C22×C4×C5C2311yescharacteristic
72C22×C4×C5C2311yescharacteristic
73C22×C4×C5C2311yescharacteristic
74C22×C4×C5C2311yescharacteristic
75C2×C22⋊D5—22no—
76C2×C22⋊D5—22no—
77C2×C22⋊D5—22no—
78C2×C22⋊D5—22no—
79C2×C22⋊D5—22no—
80C2×C22⋊D5—22no—
81C22×Dic5C2311yescharacteristic
82C22×Dic5C2311yescharacteristic
83C22×Dic5C2311yescharacteristic
84C22×Dic5C2311yescharacteristic
85C22×Dic5C2311yescharacteristic
86C22×Dic5C2311yescharacteristic
87C2×C4×D5—22no—
88C2×C4×D5—22no—
89C2×C4×D5—22no—
90C2×C4×D5—22no—
91C2×C4×D5—22no—
92C2×C4×D5—22no—
93C2×C4×D5—22no—
94C2×C4×D5—22no—
95C2×C4×D5—41no—
96C2×C4×D5—41no—
97C2×C4×D5—41no—
98C2×C4×D5—41no—
99C5×C4⋊C4—41no—
100C5×C22⋊C4—22no—
101C5×C22⋊C4—22no—
102C5×C22⋊C4—22no—
103C5×C22⋊C4—41no—
104C22⋊Dic5—22no—
105C22⋊Dic5—22no—
106C22⋊Dic5—22no—
107C22⋊Dic5—22no—
108C22⋊Dic5—41no—
109#685—22no—
110#685—22no—
111#685—22no—
112#685—22no—
113#685—22no—
114#685D414yescomplemented
115C4⋊Dic5—22no—
116#683—22no—
117#683—41no—
118C2×#123—51no—
119C2×#120—51no—
120C22×C22⋊C4—51no—
121C22×C22⋊C4—51no—
122#397—51no—
123#397—51no—
124#394—51no—
125#394D511yescharacteristic, complemented
126#386—51no—
127#386—51no—
128#386—51no—
129#385—51no—
130#381—51no—
131#380—51no—
132#380—51no—
133C23×C5—21no—
134C23×C5—21no—
135C23×C5—21no—
136C23×C5—21no—
137C23×C5—21no—
138C23×C5—21no—
139C23×C5—21no—
140C23×C5—21no—
141C23×C5—21no—
142C23×C5—21no—
143C23×C5—21no—
144C23×C5—21no—
145C23×C5—21no—
146C23×C5C2×D412yes—
147C23×C5C2411yescharacteristic
148C22×D5—21no—
149C22×D5—21no—
150C22×D5—21no—
151C22×D5—21no—
152C22×D5—21no—
153C22×D5—21no—
154C22×D5—21no—
155C22×D5—21no—
156C22×D5—28no—
157C22×D5—28no—
158C22×D5—44no—
159C22×D5—44no—
160C22×D5—44no—
161C22×D5—44no—
162C22×D5—44no—
163C22×D5C2×D412yes—
164C22×D5C2×D412yes—
165C22×D5C2×D412yes—
166C22×D5#4112yes—
167C22×D5#4112yes—
168C22×D5#4112yes—
169C5×D4—44no—
170C2×C4×C5—21no—
171C2×C4×C5—21no—
172C2×C4×C5—21no—
173C2×C4×C5—21no—
174C2×C4×C5—21no—
175C2×C4×C5—21no—
176C2×C4×C5—21no—
177C2×C4×C5—21no—
178C2×C4×C5—21no—
179C2×C4×C5—21no—
180C2×C4×C5—21no—
181C2×C4×C5—21no—
182C2×C4×C5—21no—
183C2×C4×C5—21no—
184C2×C4×C5C2×D412yes—
185C22⋊D5—44no—
186C22⋊D5—44no—
187C2×Dic5—21no—
188C2×Dic5—21no—
189C2×Dic5—21no—
190C2×Dic5—21no—
191C2×Dic5—21no—
192C2×Dic5—21no—
193C2×Dic5—21no—
194C2×Dic5—21no—
195C2×Dic5—21no—
196C2×Dic5—21no—
197C2×Dic5—21no—
198C2×Dic5—21no—
199C2×Dic5—21no—
200C2×Dic5—21no—
201C2×Dic5—21no—
202C2×Dic5—21no—
203C2×Dic5—21no—
204C2×Dic5C2×D412yes—
205C4×D5—44no—
206C4×D5—44no—
207C4×D5—44no—
208C4×D5—44no—
209C25—51no—
210C22×D4—51no—
211C22×D4—51no—
212C22×D4C2×D511yescharacteristic
213C23×C4—51no—
214C23×C4—51no—
215C23×C4—51no—
216C23×C4—51no—
217#123—104no—
218#120—104no—
219C2×C4⋊C4—51no—
220C2×C4⋊C4—51no—
221C2×C4⋊C4—51no—
222C2×C4⋊C4C2×D511yescharacteristic, complemented
223C2×C22⋊C4—51no—
224C2×C22⋊C4—51no—
225C2×C22⋊C4—51no—
226C2×C22⋊C4—51no—
227C2×C22⋊C4—51no—
228C2×C22⋊C4—51no—
229C2×C22⋊C4—51no—
230C2×C22⋊C4—51no—
231C2×C22⋊C4—51no—
232C2×C22⋊C4—51no—
233C2×C22⋊C4—51no—
234C2×C22⋊C4—104no—
235C2×C22⋊C4—104no—
236C2×C22⋊C4C2×D511yescharacteristic
237C2×C22⋊C4C2×D511yescharacteristic
238C2×C22⋊C4C2×D511yescharacteristic, complemented
239C2×C22⋊C4C2×D511yescharacteristic, complemented
240#95—51no—
241#95—51no—
242#95—51no—
243#95—51no—
244#95—51no—
245#95—51no—
246#95—51no—
247#95C2×D511yescharacteristic, complemented
248C22×C5—22no—
249C22×C5—22no—
250C22×C5—22no—
251C22×C5—41no—
252C22×C5—41no—
253C22×C5—41no—
254C22×C5—41no—
255C22×C5—41no—
256C22×C5—41no—
257C22×C5—41no—
258C22×C5—41no—
259C22×C5—41no—
260C22×C5—41no—
261C22×C5—41no—
262C22×C5C22×D411yescharacteristic
263C22×C5C22×D411yescharacteristic
264C22×C5C2×#4111yescharacteristic
265C22×C5C2×#4111yescharacteristic
266C22×C5C2×#4111yescharacteristic
267C22×C5H(5,𝔽2)11yescharacteristic
268C22×C5H(5,𝔽2)11yescharacteristic
269C2×D5—22no—
270C2×D5—22no—
271C2×D5—22no—
272C2×D5—22no—
273C2×D5—22no—
274C2×D5—22no—
275C2×D5—22no—
276C2×D5—22no—
277C2×D5—22no—
278C2×D5—22no—
279C2×D5—22no—
280C2×D5—22no—
281C2×D5—416no—
282C2×D5#12114yes—
283C2×D5#12314yescomplemented
284C4×C5—22no—
285C4×C5—41no—
286C4×C5—41no—
287C4×C5—41no—
288C4×C5—41no—
289Dic5—22no—
290Dic5—41no—
291Dic5—41no—
292Dic5—41no—
293Dic5—41no—
294Dic5—41no—
295C24—51no—
296C24—51no—
297C24—54no—
298C24—102no—
299C24—102no—
300C24—102no—
301C24—102no—
302C24—201no—
303C24—201no—
304C24C22×D511yescharacteristic
305C24C22×D511yescharacteristic
306C2×D4—22no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—102no—
310C2×D4—102no—
311C2×D4—102no—
312C2×D4—102no—
313C2×D4—102no—
314C2×D4—102no—
315C22×C4—51no—
316C22×C4—51no—
317C22×C4—51no—
318C22×C4—51no—
319C22×C4—51no—
320C22×C4—51no—
321C22×C4—102no—
322C22×C4—102no—
323C22×C4—102no—
324C22×C4—102no—
325C22×C4—102no—
326C22×C4—102no—
327C22×C4—102no—
328C22×C4—102no—
329C22×C4—201no—
330C22×C4—201no—
331C22×C4—201no—
332C22×C4—201no—
333C22×C4C22×D511yescharacteristic
334C22×C4C22×D511yescharacteristic
335C22×C4C22×D511yescharacteristic
336C22×C4C22×D511yescharacteristic
337C22×C4C22×D511yescharacteristic
338C4⋊C4—41no—
339C4⋊C4—102no—
340C4⋊C4—102no—
341C4⋊C4—201no—
342C22⋊C4—22no—
343C22⋊C4—22no—
344C22⋊C4—22no—
345C22⋊C4—41no—
346C22⋊C4—54no—
347C22⋊C4—102no—
348C22⋊C4—102no—
349C22⋊C4—102no—
350C22⋊C4—102no—
351C22⋊C4—102no—
352C22⋊C4—102no—
353C22⋊C4—102no—
354C22⋊C4—102no—
355C22⋊C4—102no—
356C22⋊C4—201no—
357C2×C5—42no—
358C2×C5—81no—
359C2×C5C2×#12111yescharacteristic
360C2×C5C2×#12311yescharacteristic
361C2×C5#53411yescharacteristic
362C2×C5#54611yescharacteristic
363C2×C5#54611yescharacteristic
364C2×C5#55111yescharacteristic
365C2×C5#55111yescharacteristic
366D5—24no—
367D5—24no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—52no—
382C23—52no—
383C23—52no—
384C23—52no—
385C23—52no—
386C23—52no—
387C23—108no—
388C23—108no—
389C23—101no—
390C23—101no—
391C23—101no—
392C23—101no—
393C23—101no—
394C23—101no—
395C23—101no—
396C23—101no—
397C23—204no—
398C23—204no—
399C23—204no—
400C23—204no—
401C23—204no—
402C23D4×D512yes—
403C23C23×D511yescharacteristic
404D4—44no—
405D4—204no—
406D4—204no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—52no—
422C2×C4—101no—
423C2×C4—101no—
424C2×C4—101no—
425C2×C4—101no—
426C2×C4—101no—
427C2×C4—101no—
428C2×C4—101no—
429C2×C4—101no—
430C2×C4—101no—
431C2×C4—101no—
432C2×C4—101no—
433C2×C4—101no—
434C2×C4—101no—
435C2×C4—101no—
436C2×C4—101no—
437C2×C4—101no—
438C2×C4—101no—
439C2×C4—204no—
440C2×C4—204no—
441C2×C4—204no—
442C2×C4—204no—
443C2×C4D4×D512yes—
444C5#241211yescharacteristic, complemented, Sylow
445C22—22no—
446C22—22no—
447C22—22no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—54no—
460C22—54no—
461C22—102no—
462C22—102no—
463C22—102no—
464C22—102no—
465C22—102no—
466C22—102no—
467C22—102no—
468C22—102no—
469C22—102no—
470C22—102no—
471C22—102no—
472C22—102no—
473C22—2016no—
474C22C2×D4×D511yescharacteristic
475C22C2×D4×D511yescharacteristic
476C22#414411yescharacteristic
477C22#414411yescharacteristic
478C22D5×#4111yescharacteristic
479C22D5×#4111yescharacteristic
480C22D5×#4111yescharacteristic
481C4—22no—
482C4—41no—
483C4—41no—
484C4—41no—
485C4—41no—
486C4—102no—
487C4—201no—
488C4—201no—
489C4—201no—
490C4—201no—
491C4—201no—
492C2—42no—
493C2—81no—
494C2—104no—
495C2—104no—
496C2#6592311yescharacteristic
497C2D5×#12111yescharacteristic
498C2#6594411yescharacteristic
499C2#6594611yescharacteristic
500C2D5×#12311yescharacteristic
501C2#6599311yescharacteristic
502C2#6601311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#394C2
prime radicalC2×#4083C2
hypercenter#394D5
hypocenterC5#2412
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #394

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#4029C2semidirect
3. C2×D5×C22⋊C4C2semidirect
4. C2×D5×C22⋊C4C2semidirect
5. C5×#394C2semidirect
6. #65511C2semidirect
7. #65278C2semidirect
8. #65276C2semidirect
9. #65245C2semidirect
10. #65245C2semidirect
11. #65245C2semidirect
12. #65242C2semidirect
13. #65241C2semidirect
14. #64956C2semidirect
15. #64953C2semidirect
24. C2×C5×C4⋊C4C22semidirect
27. C2×C5×C22⋊C4C22semidirect
28. C2×C5×C22⋊C4C22semidirect
32. C2×C22⋊Dic5C22semidirect
33. C2×C22⋊Dic5C22semidirect
41. C2×C4⋊Dic5C22semidirect
42. C2×#683C22semidirect
43. C2×#683C22semidirect
47. C5×#95C22semidirect
48. #3963C22semidirect
49. #3963C22semidirect
50. #3963C22semidirect
51. #3963C22semidirect
52. #3963C22semidirect
53. #3963C22semidirect
54. #3963C22semidirect
114. #685D4semidirect
125. #394D5semidirect
222. C2×C4⋊C4C2×D5semidirect
238. C2×C22⋊C4C2×D5semidirect
239. C2×C22⋊C4C2×D5semidirect
247. #95C2×D5semidirect
283. C2×D5#123semidirect
444. C5#2412semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#40832
C2×#40292
C2×D5×C22⋊C42
C5×#3942
#655112
#652782
#652762
#652452
#652422
#652412
#649562
#649532
#24125